models of link capacity distribution in isp's router-level topology
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International Journal of Computer Networks & Communications (IJCNC) Vol.3, No.5, Sep 2011
DOI : 10.5121/ijcnc.2011.3515 205
MODELS OF LINK CAPACITY
DISTRIBUTION IN ISP’S ROUTER-LEVEL
TOPOLOGY Takahiro Hirayama, Shin’ichi Arakawa, Shigehiro Hosoki and Masayuki Murata
Graduate School of Information Science and Technology,Osaka University, Osaka, Japan
{t-hirayama, arakawa, s-hosoki, murata}@ist.osaka-u.ac.jp
ABSTRACT
Modeling the Internet is vital for network researches. Recent measurement studies on the Internet topology show that the degree distribution obeys the power-law distribution. However, only the degree
distribution does not determine the performance of network control methods. As previous studies have
shown, one of important factors to characterize the performance of network control methods in the
Internet is a structure of topologies. However, other characteristics, which are even more important, are
link capacities and node processing capacities of the network because these characteristics are
particular to communication networks. In this paper, we investigate how to model the link capacity in the
router-level topologies. We first reveal that the link capacity distribution of ISP’s backbone network in
Japan obeys a power-law with exponent -1.1. To clarify the reason for the link capacity distribution, we
evaluate throughput of networks having various kinds of link capacity distributions. Our numerical
results show that the network with power-law link capacity distribution can accommodate much more
traffic than the network with exponential distributions of link capacities.
K EYWORDS
Power-law, Router-level Topology, Link Capacity, Modeling the Internet, Traffic Demand Matrix,
Lognormal Distribution
1. INTRODUCTION
Modeling the Internet is one of important issues to evaluate network control methods [1]. Since
the performance of network control methods strongly depends on Internet topology and
capacity of links, a proper network model is necessary to show effectiveness of those methods.
Therefore, it is important to reveal characteristics of a communication network and origin of
them.
Measurement studies on the Internet topology show that the degree distribution obeys the
power-law [2], [3]. That is, the existence probability P(k) of node with K out-going linksapproximates to K
-r (r is constant). There are many modeling methods for the power-law
topology [4-6]. Reference [5] presents FKP model for a topology generation model that
consider technical constraints of router-level topologies. However, the topology is much
different from the actual ISP topologies [7]. In Ref. [6], the authors enumerate several
topologies having the same degree distribution. Then, they evaluate the amount of traffic that
the network can accommodate in each topology under the constraint of node processing
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capacities. The results indicate that the network throughput highly depends on the structure of
topologies even though they have the same degree distribution.
Many researches about the modeling method of router-level topology have been discussed
based on the characteristic that the degree distribution obeys the power-law. In theseresearches, there was an assumption that link capacities can be ignored, or they are identical.
However, the performance of network strongly depends on link capacities because link
capacities are characteristics particular to communication networks.
In this paper, we investigate the characteristic of link capacities in router-level topology. One of
our main motivations in this paper is to investigate the modeling methodology for realistic ISP
networks. For this purpose, characteristics particular to communication networks other than the
degree distribution are important. First, we investigate the link capacity distribution in a
Japanese ISP network using the capacity information presented in Ref. [8]. Results show that
the link capacity distribution also obeys the power-law with exponent -1.11. Second, to reveal
the reason why the link capacity distribution obeys the power-law, we investigate the network
performance under the constraint of link capacities. We generate three distributions of link
capacities; power-law, exponential and identical. Then we assign the link capacities to ISP
topologies, and compare the network performance. Results show that topologies with power-
law link capacity distribution can accommodate more traffic than that with other two
distributions. We also reveal that it is insufficient to consider only the constraint of nodeprocessing capacities, even though it is discussed in Ref. [6], in evaluating the network
performance.
This paper is organized as follows. We introduce related work in Section 2. In Section 3, we
investigate link capacity distribution in a Japanese ISP network. In Section 4, we explain about
topology generation models and topologies which are used for evaluation. In Section 5, we
investigate the difference of network performance by three distributions of link capacities.
Section 6 concludes this paper.
2. RELATED WORK
There are relatively few studies on modeling router-level Internet topology [4–6, 9, 10].
Heckmann et. al. [9] present parameter settings for topology generator tools such as BRITE,
TIERS, and GT-ITM to construct POP (point-of-presence)-level ISP topologies. Topology
generation methods that consider technical constraints of router-level topologies are discussed
in [5, 6]. Reference [5] presents FKP model where a newly added node connects with existingnodes that minimize the sum of Euclidian distance between the nodes and logical distance from
the existing node. The authors demonstrate that the degree distribution obeys the power-law
with appropriate parameter settings. However, the topology generated by the model has too
many nodes whose degree is one [11], and thus the topology is much different from the actual
ISP topologies [7]. This is because ISP topologies are designed with technological, geographic,and business constraints [12, 13]. In Ref. [6], the authors enumerate several topologies having
the same degree distribution. Then, they evaluate the amount of traffic that the network can
accommodate in each topology under the constraint of node processing capacities. The results
indicate that the network throughput highly depends on the structure of topologies even though
they have the same degree distribution. With the constraint of node processing capacities, aproduct of node degree and its corresponding link capacities is bounded. Thus, for maximize
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the traffic that the network can accommodate, non-hub nodes having small number of links
connects with higher-capacity links, and hub-nodes having many links connects with lower-
capacity links.
Moreover, many researchers have discussed about capacity distribution in power-lawtopologies [14–16]. Zhang et al. mentioned that node capacity distribution based on node
betweenness centrality is appropriate to enhance network capacity [14]. Betweenness centrality
is defined as the number of shortest paths that pass through the nodes or edges. In Ref. [15],
Xia et al. assigned optimal node capacity with an optimization problem to maximize the system
performance and the utilization ratio of capacity. This method provides a similar capacity
distribution to that is allocated to be proportional to betweenness centrality of nodes in BA topologies. These researchers assumed that link capacity is unlimited. However, actual
communication systems like the Internet have heterogeneous link capacity distribution. In Ref.
[16], Ling et al. discussed about link capacity distribution in BA topologies. They claimed that
link capacity distribution based on the product of both end nodes’ degree maximizes network
throughput. In these papers, model-based topologies are concerned. Model-based topologies
Figure 1: Node Processing Capacity Distribution in the IIJ Network
Figure 2: Link Processing Capacity Distribution in the IIJ Network
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have different properties from ISP topologies’. So, we evaluate the relationship between link
capacity distribution and network throughput in ISP topologies.
3. LINK CAPACITY DISTRIBUTION IN ISP ROUTER-LEVELTOPOLOGY
Ref. [8] investigates node processing capacities of a commercial ISP network in Japan. The
author uses a disclosed information of link capacities in IIJ (Internet Initiative Japan) [17]. Inthe paper, a node processing capacity is defined as the sum of link capacity connected to the
node. Figure 1, which was also presented in Ref. [8], shows the rank of node processing
capacities in 2002. The vertical axis represents the node processing capacity and the horizontal
axis represents the rank of that node. We observe that node processing capacity distribution
obeys the power-law; its exponent is nearly -2.6.
The result is interesting in that the power-law relationship again appears in the node processing
capacity. However, one question arises whether the link capacity distribution follows the
power-law or not. To answer this, we show the link capacity distribution in Figure 2; the resultshows that the exponent of the link capacity distribution (for 40 higher-ranking) is -1, i.e., the
distribution obeys the zipf’s law. The figure is obtained by using the information on IIJ
backbone network in 2002, thus it is uncertain that other ISP topologies have the observation
that the link capacity distribution obeys the zipf’s law. However, we believe that theobservation is general because we can easily derive it by the non-blocking configuration of
routers. Supposing that there is one uplink port with the capacity of in a router and there are α
numbers of links, then if the router to be non-blocking, the capacity of each link should be X / α.With this case, the exponent of link capacity distribution becomes -1. Base of this discussion, in
the next section, we investigate the performance of networks when the link capacity distribution
obeys the power-law with exponent -1.
4. TOPOLOGIES FOR EVALUATION4.1. ISP Topologies
We prepare two ISP topologies; AT&T and Sprint [18]. AT&T topology has 523 nodes and
1304 links, and Sprint topology has 467 nodes and 1280 links. These topologies measured
using the Rocketfuel tool [19].
4.2. Abeline-inspired Topology
As a router-level topology, we prepare the Abilene-inspired topology that is also evaluated in
Ref. [6]. This topology is based on Abilene Network used for higher education. The Abilene-
inspired topology has 869 nodes and 877 links.
4.3. BA Topology
Barabasi and Albert proposed a BA model to generate topologies having a power-law degree
distribution [4]. We call the BA topology as the topology generated by BA model. The BA
model is characterized by two features: Incremental Growth and Preferential Attachment, The
model starts with a topology with a small number of nodes and works as follows:
Step1 Make an initial topology that has mo nodes.
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Step2 Incremental Growth: Add a new node at each time step.
Step3 Preferential Attachment: Connect the new node to m different nodes chosen with
probability :
(1)
Where k i is the degree of node i .
To follow this process, the topology with its degree distribution obeys the power-law is
generated. In this paper, we generate BA topology to be the same number of nodes and links in
Abilene-inspired topology. At first we use BA model with m0=1, and generate 869 nodes and
868 links topology. Then, nine links are added based on rule of Step3. The topology with 869
nodes and 877 links is generated.
4.4. FKP Topology
The FKP model proposed by Fabrikant et al. [5] revealed that the power-law property of degreedistribution can still be obtained by minimizing “distance” metrics. This model does not use
preferential attachment to add links, and instead uses minimization-based link attachment.
More specifically, the FKP model works as follows. Each new node arrives at randomly in the
Euclidean space {0,1}2
. After arriving at new node i, the FKP model calculates the following
equation for each node, j, already existing in the network: β .wij+loj where wij is the Euclidean
distance (i.e., physical distance) between nodes and , and is the hop-counts distance between
node and a pre-specified “root” node (node 0). β is a parameter that weights the importance of physical distance. If β has a lower value, each node tries to connect to higher degree nodes; β =0
is an extreme scenario that creates a star-topology. If β has a higher value, each node tries to
connect their nearest nodes. A topology with high a β is shown to behaves like an ER topology.
The power-law property of the degree distribution appears at a
Figure 3: Normalized Link Capacity Distributions
moderate value of β value. Here, there are several hub-nodes in each region, and the hub-nodes
form a power-law. However, it is mentioned that the number of node whose degree is one is
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much more than that of ISP topology [6] and the degree distribution is different from that of
ISP topology [18].
In this paper, we generate FKP topology to be the same number of nodes and links in Abilene-
inspired topology. For this purpose, we modify the FKP model; we first select nine nodesrandomly before starting the topology generation and when each of these nine nodes is added,
we generate two links between itself and a node satisfying the equation ( β .wij+loj) , and
another next node satisfying the equation. At last, a topology which has 869 nodes and 877
links is generated. )
5. EVALUATION ON LINK CAPACITY DISTRIBUTION
In Section 3, we show that the link capacity distribution of Japanese ISP obeys the power-law.
For revealing the reason that link capacity distribution obeys power-law, we investigate
network performance under the constraint of link capacities. In this paper, we define the
network performance as the amount of traffic that the topology can accommodate. For
comparison purpose, we generate three distributions of link capacities; power-law, exponential,and identical where all link capacities are identical. Then, the amount of traffic that the
topology can accommodate for each distribution is compared.
5.1. Link Capacity Distribution
According to the observation in Section 3, we generate link capacity distribution that obeys
power-law with exponent the -1. We first prepare m links with the capacity X [bps]. Then, we
prepare α X m links each of which has X / α link capacity. By setting m ← α X m and,
X ← X / α , we repeatedly prepare links. With this case, the exponent of link capacity
distribution becomes -1.
For comparison purpose, we generate other two link capacity distributions that obeyexponential and identical. Figure 3 shows obtained distributions. The vertical axis represents
the link capacity and horizontal axis represents the rank of that link. Note that we normalize theexponential and the identical distributions such that the sum of link capacities in each
distribution is equaled.
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Figure 4: The Amount of Traffic that AT&T
Topology can Accommodate
Figure 5: The Amount of Traffic that Sprint
Topology can Accommodate
Figure 6: The Amount of Traffic that BATopology can Accommodate
Figure 7: The Amount of Traffic that FKPTopology can Accommodate
5.2. Assigning Link Capacity
We then assign link capacity to each link in the topology. In this paper, we assume that the link
capacity is assigned based on the edge betweenness centrality. More exactly, we assign link
capacities in a descending-order to link with the edge betweenness centrality. The edge
betweenness centrality is defined as the number of minimum hop paths that pass through thelink. That is, the edge betweenness centrality represents the amount of contribution of a link to
the communication in the network.
5.3. Evaluation
We evaluate the amount of traffic that each topology can accommodate under the constraint of
link capacities. The traffic demand is generated by log-normal distribution since the distribution
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is known to be suitable for modeling the traffic demand [20]. We change the variance of the
log-normal distribution from 0.1 to 10.0 with the step 0.1. The traffic demand is generated for
each variance and sum of the traffic demands is normalized to 1. A minimum hop routing is
applied to select the route of the traffic. Note that if there are several paths that have the samenumber of hops, one of them is selected randomly. The result is shown in Figure 4-8. The
vertical axis represents the amount of traffic that the topology can accommodate and horizontal
axis represents the variance of traffic demand. In these figures, 95% confidence interval is also
presented.
In ISP topologies (Figure 4 and Figure 5), when the variance of traffic demand is less than 2,
the topology with the power-law link capacity distribution can accommodate more traffic than
Figure 8: The Amount of Traffic that Abiline-inspired Topology can Accommodate
Figure 9: Relationship between Node Degree and Node Processing Capacity
topologies with other distributions. In Ref. [20], the variance of log-normal distribution is
nearly 1 for fitting the observed traffic in real ISPs. Thus, in the realistic situation, the power-
law link capacity distribution makes the topology more efficient. The topology with the power-
law distribution accommodates the largest traffic among three distributions even though the
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variance of traffic demand changes around 1.0. However, the advantage of the power-law
distribution decreases and diminishes when the variance is too large since elephant traffic
appears in the traffic matrix. The same results are also observed in Figure 6, 7, 8. From the
results of these figures, the power-law link capacity distribution can accommodates largeramount of the traffic than that of other distributions.
5.4. Effects of Node Processing Constraint
In Ref. [6], it is mentioned that the number of links connected to a router affects capacities of
link. Figure 9 shows a relationship between node degree and node processing capacity. Due to
the constraint of router technology, a product of node degree and its link capacities is bounded
by the node processing capacity. That is, a router having few links can accommodate large
capacity links, and a router having small capacity of links can accommodate many links. This
section examines the difference of network performance between two constraints for the link
capacities and the node processing capacities. We apply the relation in Figure 9 to the Sprint
topology and Abilene-inspired topology. Figure 10 and Figure 11 show the amount of traffic
that the topology can accommodate under the constraint of node processing capacities. For the
comparison purpose, we again show the result of previous section in (a) of each figure and the
result under the constraint of node processing capacities in (b) of each figure.
(a) The Case of Link Capacity Constraints (b) The Case of Node Capacity Constraints
Figure 10: The Network Performance of Sprint Topology
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(a) The Case of Link Capacity Constraints (b) The Case of Node Capacity Constraints
Figure 11: The Network Performance of Abiline-inspired Topology
Comparison between (a) and (b) of Figure 10 and Figure 11 shows the difference of amount of
the traffic that the topology can accommodate with each constraint; the link capacities and the
node processing capacities. These figures have a common tendency that amount of the traffic
that the topology can accommodate gets smaller as the variance of traffic demand becomes
larger. In other three topologies (BA topology, FKP topology and Abilene-inspired topology),
the same tendency is observed. However, the amount of traffic that the topology can
accommodate under the constraint of link capacities is strongly smaller than that under the
constraint of node processing capacities. Thus, evaluation of network performance only theconstraint of node processing capacities by its degree is inadequate for network evaluations.
6. CONCLUSION
Evaluation of the network control method needs proper model of the communication network.
In this paper, we focused on link capacity in ISP router-level topology. First, we investigate
link capacity distribution in Japanese ISP backbone network. Results show that link capacity
distribution in ISP obeys the power-law. Next, for revealing the reason that link capacity
distribution obeys power-law, we compared network performance with three link capacitydistributions; power-law, exponential and identical. Results show that power-law link capacity
distribution can accommodate large amount of the traffic. Moreover, under the constraint of
only node processing capacities by its degree, amount of the traffic that the network canaccommodate is over estimated. Thus to take characteristic that link capacity distribution obeys
power-law into the modeling of router-level topology is important to evaluate the network control method properly.
ACKNOWLEDGEMENT
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This research was supported in part by “Global COE (Centers of Excellence) Program” of the
Ministry of Education, Culture, Sports, Science and Technology, Japan and “Grant-in-Aid for
Scientific Research (A) 21240004” of the Japan Society for the Promotion of Science (JSPS) in
Japan.
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Authors
Takahiro Hirayama received a M.S. degree from Osaka University in 2010, where he is
currently a doctoral student in Information Science and Technology. His research interest
includes complex networks.
Shin’ichi Arakawa received M.E. and D.E. degrees in informatics and mathematical sciencefrom Osaka University in 2000 and 2003. He is currently an Assistant Professor at the Graduate
School of Information Science and Technology, Osaka University. His research interests
include optical networks and complex networks. He is a Member of IEEE.
Shigehiro Hosoki received a M.S. degree from Osaka University in 2011. His research interest
includes complex networks.
Masayuki Murata received M.E. and D.E. degrees in information and computer sciences from
Osaka University in 1984 and 1988. In April 1984, he joined the Tokyo Research Laboratory
IBM Japan as a Researcher. From September 1987 to January 1989, he was an Assistant
Professor with the Computation Center, Osaka University. In February 1989, he moved to theDepartment of Information and Computer Sciences, Faculty of Engineering Science, Osaka
University. From 1992 to 1999, he was an Associate Professor with the Graduate School of
Engineering Science, Osaka University, and since April 1999, he has been a Professor. He
moved to the Graduate School of Information Science and Technology, Osaka University in
April 2004. He has published more than 300 papers in international and domestic journals and
conferences. His research interests include computer communication networks, performancemodeling, and evaluation. He is a Member of IEEE, the Association for Computing Machinery
(ACM), The Internet Society, and IPSJ.